Chap. VIII] MISCELLANEOUS RESULTS ON CONGEUENCES. 261 
E. Landau^'^ proved that, if /(x) =0 is an equation with integral coeffi- 
cients and at least one root of odd multiplicity, there exist an infinitude of 
primes p = 4:n — l such that /(a;) = (mod p) has a root. 
R. D. von Sterneck^^^ found the number of combinations of the ith. class 
(with or without repetition) of the numbers prime to p of a complete set of 
residues modulo p^ whose sum is congruent to a given integer modulo p^, 
p being a prime. 
E. Piccioli^^'* gave known theorems on adding and multiplying con- 
gruences. 
C. Jordan^^^ found the number of sets of integers aik for which the 
determinant |aa-| of order n is congruent to a given integer modulo M. 
C. Krediet^^^ gave theorems on congruences of degree n for a prime 
modulus analogous to those for an algebraic equation of degree n, including 
the question of multiple roots. The determination of roots is often sim- 
plified by seeking first the roots which are quadratic residues and then 
those which are non-residues. The exposition is not clear or simple. 
G. Rados"^ proved that, if p is a prime, 
fix) = ao^^-^+ • • ■ +«p-2= 0, g{x) = hx^'-^-i- . . . +bp_2= (mod p) 
have a common root if and only if each Ri=0 (mod p), where 
*(w) =Rou''-^+Riu''-^-\- . . . +Rp-i 
aou+bo aiU+bi ... ap_2W+6p_2 
aiu+bi a2U-\-b2 . . . aoU-{-bo 
ap_2U+bp-2 aou+bo ... ttp-s^+^p-s 
For ^=/', let ^(u) become DoU^~^-\- . . . +i)p_2; thus/(a:)^0 (mod p) has a 
multiple root if and only if each A=0 (mod p). Each of these theorems 
is extended to three congruences. Finally, if f(x) and f'(x) are relatively 
prime algebraically, there is only a finite number of primes p for which the 
number of roots of /= (mod p'') exceeds the degree of /. 
G. Frattini^^^ proved that if p and q are primes, q a divisor of p — 1, 
every homogeneous symmetric congruence in q variables is solvable modulo 
p by values of the variables distinct from each other and from zero except 
when the degree of the congruence is divisible by q. 
C. Grotzsch^'^ noted that if a is a root of a^'^^a (mod p), where a is prime 
to p, then x=a (mod p^ — p) is a root, and proved that if d is the g. c. d. of 
ind a and p — 1 and if ind a>0, it has exactly 
^''^Handbuch . . .Verteilung der Primzahlen, 1, 1909, 440. 
^"Sitzungsber. Ak. Wiss. Wien (Math.), 118, 1909, Ila, 119-132. 
'^"11 Pitagora, Palermo, 16, 1909-10, 125-7. 
"5Jour. de Math., (6), 7, 1911, 409-416. 
'^^Wiskundig Tijdschrift, Haarlem, 7, 1911, 193-202 (Dutch). 
i"Ami. sc. ecole norm, sup., (3), 30, 1913, 395-412. 
"speriodico di Mat., 29, 1913, 49-53. 
"'Archiv Math. Phys., (3), 22, 1914, 49-53. 
